{"id":719,"date":"2024-04-20T16:48:22","date_gmt":"2024-04-20T16:48:22","guid":{"rendered":"https:\/\/gbelentzas.sites.sch.gr\/?p=719"},"modified":"2024-05-01T07:45:48","modified_gmt":"2024-05-01T07:45:48","slug":"%ce%b1%ce%b8","status":"publish","type":"post","link":"https:\/\/gbelentzas.sites.sch.gr\/?p=719","title":{"rendered":"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd – \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7)"},"content":{"rendered":"

\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 19. <\/strong>\u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7<\/span> \"\textcolor{blue}{f }\" \u00a0\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1<\/span> \"\textcolor{blue}{(\alpha,\beta), }\" \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5<\/span> \"\textcolor{blue}{ x_0,}\" \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03cc\u03bc\u03c9\u03c2 \u03b7<\/span> \"\textcolor{blue}{f }\" \u00a0\u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2.<\/span><\/p>\n

\u0391\u03bd<\/span> \"\textcolor{blue}{ f'(x) > 0}\" \u03c3\u03c4\u03bf<\/span> \"\textcolor{blue}{ (\alpha,x_0)}\" \u03ba\u03b1\u03b9<\/span>\u00a0 \"\textcolor{blue}{ f'(x) <0}\" \u00a0\u03c3\u03c4\u03bf<\/span> \"\textcolor{blue}{ (x_0,\beta),}\" \u03c4\u03cc\u03c4\u03b5 \u03bd\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03bf<\/span> \"\textcolor{blue}{ f(x_0)}\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03b7\u03c2<\/span> \"\textcolor{blue}{ f}\".<\/p>\n

(\u0397\u039c. 2012, \u0397\u039c, 2019)<\/em><\/span><\/p>\n

\n

\u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \"f\" \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \"(\alpha,\beta),\" \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \"x_0,\" \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03cc\u03bc\u03c9\u03c2 \u03b7 \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2, \u03bc\u03b5 \"f '(x) > 0\" \u03c3\u03c4\u03bf \"(\alpha, x_0)\" \u03ba\u03b1\u03b9 \"f'(x) < 0\" \u03c3\u03c4\u03bf \"(x_0 , \beta)\"<\/p>\n

\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \"f'(x) > 0\" \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \"x \in (\alpha, x_0)\" \u03ba\u03b1\u03b9 \u03b7 \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \"x_0,\" \u03b7 \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \"(\alpha,x_0].\" \u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5<\/p>\n

\"f(x) \leq f(x_0)\", \u00a0\u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \"x \in (\alpha, x_0]. \quad (1)\"<\/p>\n

\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \"f'(x) < 0\" \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \"x \in (x_0, \beta)\" \u03ba\u03b1\u03b9 \u03b7 \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \"x_0,\" \u03b7 \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \"[x_0,\beta).\" \u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5:<\/p>\n

\"f(x) \leq f(x_0)\", \u00a0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u00a0\"x \in [x_0, \beta). \quad (2)\"<\/p>\n

\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03c9\u03bd (1) \u00a0\u03ba\u03b1\u03b9 (2), \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9:<\/p>\n

\"f(x) \leq f(x_0)\", \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \"x \in (\alpha, \beta)\"<\/p>\n

\u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03c4\u03bf \"f(x_0)\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03b7\u03c2 \"f\" \u03c3\u03c4\u03bf \"(\alpha,\beta)\" \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03b1\u03c5\u03c4\u03ae\u03c2.<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 19. \u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u00a0\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03cc\u03bc\u03c9\u03c2 \u03b7 \u00a0\u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2. \u0391\u03bd \u03c3\u03c4\u03bf \u03ba\u03b1\u03b9\u00a0 \u00a0\u03c3\u03c4\u03bf \u03c4\u03cc\u03c4\u03b5 \u03bd\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03b7\u03c2 . (\u0397\u039c. 2012, \u0397\u039c, 2019) \u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-719","post","type-post","status-publish","format-standard","hentry","category-13"],"featured_image_src":null,"author_info":{"display_name":"admin","author_link":"https:\/\/gbelentzas.sites.sch.gr\/?author=1"},"yoast_head":"\n\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd - \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7) - MathsEdu.gr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/gbelentzas.sites.sch.gr\/?p=719\" \/>\n<meta property=\"og:locale\" content=\"el_GR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd - \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7) - MathsEdu.gr\" \/>\n<meta property=\"og:description\" content=\"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 19. \u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u00a0\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \u03c3\u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03cc\u03bc\u03c9\u03c2 \u03b7 \u00a0\u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2. \u0391\u03bd \u03c3\u03c4\u03bf \u03ba\u03b1\u03b9\u00a0 \u00a0\u03c3\u03c4\u03bf \u03c4\u03cc\u03c4\u03b5 \u03bd\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03b7\u03c2 . (\u0397\u039c. 2012, \u0397\u039c, 2019) \u0388\u03c3\u03c4\u03c9 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3’ \u03ad\u03bd\u03b1 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \u03bc\u03b5 \u03b5\u03be\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03af\u03c3\u03c9\u03c2 \u03ad\u03bd\u03b1...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/gbelentzas.sites.sch.gr\/?p=719\" \/>\n<meta property=\"og:site_name\" content=\"MathsEdu.gr\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MathsEdu.gr\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-20T16:48:22+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-01T07:45:48+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u03a3\u03c5\u03bd\u03c4\u03ac\u03c7\u03b8\u03b7\u03ba\u03b5 \u03b1\u03c0\u03cc\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u0395\u03ba\u03c4\u03b9\u03bc\u03ce\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03b1\u03bd\u03ac\u03b3\u03bd\u03c9\u03c3\u03b7\u03c2\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 \u03bb\u03b5\u03c0\u03c4\u03cc\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719\"},\"author\":{\"name\":\"admin\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/#\\\/schema\\\/person\\\/511115b6820d1a71ade6791721485e15\"},\"headline\":\"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd – \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7)\",\"datePublished\":\"2024-04-20T16:48:22+00:00\",\"dateModified\":\"2024-05-01T07:45:48+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719\"},\"wordCount\":115,\"commentCount\":0,\"articleSection\":[\"\u0391\u03a0\u039f\u0394\u0395\u0399\u039e\u0395\u0399\u03a3\"],\"inLanguage\":\"el\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719\",\"url\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=719\",\"name\":\"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ae\u03c4\u03c9\u03bd - 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